zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Stability of the positive point of equilibrium of Nicholson’s blowflies equation with stochastic perturbations: numerical analysis. (English) Zbl 1179.60039
Summary: Known Nicholson’s blowflies equation (which is one of the most important models in ecology) with stochastic perturbations is considered. Stability of the positive (nontrivial) point of equilibrium of this equation and also a capability of its discrete analogue to preserve stability properties of the original differential equation are studied. For this purpose, the considered equation is centered around the positive equilibrium and linearized. Asymptotic mean square stability of the linear part of the considered equation is used to verify stability in probability of nonlinear origin equation. From known previous results connected with B. Kolmanovskii and L. Shaikhet, general method of Lyapunov functionals construction, necessary and sufficient condition of stability in the mean square sense in the continuous case and necessary and sufficient conditions for the discrete case are deduced. Stability conditions for the discrete analogue allow to determinate an admissible step of discretization for numerical simulation of solution trajectories. The trajectories of stable and unstable solutions of considered equations are simulated numerically in the deterministic and the stochastic cases for different values of the parameters and of the initial data. Numerous graphical illustrations of stability regions and solution trajectories are plotted.

MSC:
60H10Stochastic ordinary differential equations
34F05ODE with randomness
92D40Ecology
WorldCat.org
Full Text: DOI EuDML
References:
[1] X. Ding and W. Li, “Stability and bifurcation of numerical discretization Nicholson blowflies equation with delay,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 19413, 12 pages, 2006. · Zbl 1106.92065 · doi:10.1155/DDNS/2006/19413 · eudml:126962
[2] Q. X. Feng and J. R. Yan, “Global attractivity and oscillation in a kind of Nicholson/s blowflies,” Journal of Biomathematics, vol. 17, no. 1, pp. 21-26, 2002.
[3] W. S. Gurney, S. P. Blytne, and R. M. Nisbet, “Nicholson/s blowflies revisited,” Nature, vol. 287, pp. 17-21, 1980. · doi:10.1038/287017a0
[4] V. Lj. Kocić and G. Ladas, “Oscillation and global attractivity in a discrete model of Nicholson/s blowflies,” Applicable Analysis, vol. 38, no. 1-2, pp. 21-31, 1990. · Zbl 0715.39003 · doi:10.1080/00036819008839952
[5] M. R. S. Kulenović, G. Ladas, and Y. G. Sficas, “Global attractivity in Nicholson/s blowflies,” Applicable Analysis, vol. 43, no. 1-2, pp. 109-124, 1992. · Zbl 0754.34078 · doi:10.1080/00036819208840055
[6] J. Li, “Global attractivity in Nicholson/s blowflies,” Applied Mathematics. Series B, vol. 11, no. 4, pp. 425-434, 1996. · Zbl 0867.34042 · doi:10.1007/BF02662882
[7] J Li, “Global attractivity in a discrete model of Nicholson/s blowflies,” Annals of Differential Equations, vol. 12, no. 2, pp. 173-182, 1996. · Zbl 0855.34092
[8] M. Li and J. Yan, “Oscillation and global attractivity of generalized Nicholson/s blowfly model,” in Differential Equations and Computational Simulations (Chengdu, 1999), pp. 196-201, World Scientific, River Edge, NJ, USA, 2000. · Zbl 0958.39013
[9] J. W.-H. So and J. S. Yu, “Global attractivity and uniform persistence in Nicholson/s blowflies,” Differential Equations and Dynamical Systems, vol. 2, no. 1, pp. 11-18, 1994. · Zbl 0869.34056
[10] J. W.-H. So and J. S. Yu, “On the stability and uniform persistence of a discrete model of Nicholson/s blowflies,” Journal of Mathematical Analysis and Applications, vol. 193, no. 1, pp. 233-244, 1995. · Zbl 0834.39009 · doi:10.1006/jmaa.1995.1231
[11] J. Wei and M. Y. Li, “Hopf bifurcation analysis in a delayed Nicholson blowflies equation,” Nonlinear Analysis, vol. 60, no. 7, pp. 1351-1367, 2005. · Zbl 1144.34373 · doi:10.1016/j.na.2003.04.002
[12] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations: With Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1991. · Zbl 0780.34048
[13] M. R. S. Kulenović and G. Ladas, “Linearized oscillations in population dynamics,” Bulletin of Mathematical Biology, vol. 49, no. 5, pp. 615-627, 1987. · Zbl 0634.92013 · doi:10.1007/BF02460139
[14] M. R. S. Kulenović, G. Ladas, and Y. G. Sficas, “Global attractivity in population dynamics,” Computers & Mathematics with Applications, vol. 18, no. 10-11, pp. 925-928, 1989. · Zbl 0686.92019 · doi:10.1016/0898-1221(89)90010-2
[15] E. Beretta, V. Kolmanovskiĭ, and L. Shaikhet, “Stability of epidemic model with time delays influenced by stochastic perturbations,” Mathematics and Computers in Simulation, vol. 45, no. 3-4, pp. 269-277, 1998. · Zbl 1017.92504 · doi:10.1016/S0378-4754(97)00106-7
[16] B. Paternoster and L. Shaikhet, “About stability of nonlinear stochastic difference equations,” Applied Mathematics Letters, vol. 13, no. 5, pp. 27-32, 2000. · Zbl 0959.60056 · doi:10.1016/S0893-9659(00)00029-X
[17] L. Shaikhet, “Stability in probability of nonlinear stochastic systems with delay,” Matematicheskie Zametki, vol. 57, no. 1, pp. 142-146, 1995 (Russian), translation in Mathematical Notes, vol. 57, no. 1, pp. 103-106, 1995. · Zbl 0843.93086 · doi:10.1007/BF02309404
[18] L. Shaikhet, “Stability in probability of nonlinear stochastic hereditary systems,” Dynamic Systems and Applications, vol. 4, no. 2, pp. 199-204, 1995. · Zbl 0831.60075
[19] L. Shaikhet, “Stability of predator-prey model with aftereffect by stochastic perturbation,” Stability and Control: Theory and Applications, vol. 1, no. 1, pp. 3-13, 1998.
[20] V. B. Kolmanovskiĭ and L. Shaikhet, “New results in stability theory for stochastic functional-differential equations (SFDEs) and their applications,” in Proceedings of Dynamic Systems and Applications, vol. 1, pp. 167-171, Dynamic, Atlanta, Ga, USA, 1994. · Zbl 0811.34062
[21] V. B. Kolmanovskiĭ and L. Shaikhet, “A method for constructing Lyapunov functionals for stochastic differential equations of neutral type,” Differentsialniye Uravneniya, vol. 31, no. 11, pp. 1851-1857, 1995 (Russian), translation in Differential Equations, vol. 31, no. 11, pp. 1819-1825, 1995. · Zbl 0869.93052
[22] V. B. Kolmanovskiĭ and L. Shaikhet, “General method of Lyapunov functionals construction for stability investigation of stochastic difference equations,” in Dynamical Systems and Applications, vol. 4 of World Sci. Ser. Appl. Anal., pp. 397-439, World Scientific, River Edge, NJ, USA, 1995. · Zbl 0846.93083
[23] V. Kolmanovskiĭ and L. Shaikhet, “Some peculiarities of the general method of Lyapunov functionals construction,” Applied Mathematics Letters, vol. 15, no. 3, pp. 355-360, 2002. · Zbl 1015.39001 · doi:10.1016/S0893-9659(01)00143-4
[24] V. Kolmanovskiĭ and L. Shaikhet, “Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results,” Mathematical and Computer Modelling, vol. 36, no. 6, pp. 691-716, 2002. · Zbl 1029.93057 · doi:10.1016/S0895-7177(02)00168-1
[25] V. Kolmanovskiĭ and L. Shaikhet, “About one application of the general method of Lyapunov functionals construction,” International Journal of Robust and Nonlinear Control, vol. 13, no. 9, pp. 805-818, 2003. · Zbl 1075.93017 · doi:10.1002/rnc.846
[26] L. Shaikhet, “Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems,” Theory of Stochastic Processes, vol. 2(18), no. 1-2, pp. 248-259, 1996. · Zbl 0893.60029
[27] \uI. \BI. Gīkhman and A. V. Skorokhod, The Theory of Stochastic Processes I, vol. 210 of Die Grundlehren der mathematischen Wissenschaften, Springer, Berlin, Germany, 1974. · Zbl 0291.60019
[28] \uI. \BI. Gīkhman and A. V. Skorokhod, The Theory of Stochastic Processes II, vol. 218 of Grundlehren der mathematischen Wissenschaften, Springer, Berlin, Germany, 1975. · Zbl 0305.60027
[29] \uI. \BI. Gīkhman and A. V. Skorokhod, The Theory of Stochastic Processes III, vol. 232 of Grundlehren der mathematischen Wissenschaften, Springer, Berlin, Germany, 1979. · Zbl 0404.60061
[30] V. B. Kolmanovskiĭ and V. R. Nosov, Stability of Functional-Differential Equations, vol. 180 of Mathematics in Science and Engineering, Academic Press, London, UK, 1986. · Zbl 0593.34070
[31] L. Shaikhet, “Necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations,” Applied Mathematics Letters, vol. 10, no. 3, pp. 111-115, 1997. · Zbl 0883.39005 · doi:10.1016/S0893-9659(97)00045-1
[32] M. Bandyopadhyay and J. Chattopadhyay, “Ratio-dependent predator-prey model: effect of environmental fluctuation and stability,” Nonlinearity, vol. 18, no. 2, pp. 913-936, 2005. · Zbl 1078.34035 · doi:10.1088/0951-7715/18/2/022
[33] M. Carletti, “On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment,” Mathematical Biosciences, vol. 175, no. 2, pp. 117-131, 2002. · Zbl 0987.92027 · doi:10.1016/S0025-5564(01)00089-X
[34] S. I. Resnick, “Adventures in Stochastic Processes,” Birkhäuser, Boston, Mass, USA, 1992. · Zbl 0762.60002